Bring 4 L of water to a rolling boil in a large stockpot. Add a pinch of salt. Grasp each live lobster from the back and carefully lower into the boiling water. Cook 8–10 minutes for 1 1/4–1 1/2 lb lobsters until shells are bright red and meat is opaque. Remove lobsters with tongs and transfer to an ice bath briefly to stop cooking.

Reserve meat and shells

When cool enough to handle, crack lobster shells and remove tail and claw meat. Reserve about 1 cup of bite-size pieces refrigerated for finishing. Keep shells (bodies, claws, tail shells) for stock.

Sauté shells and vegetables

In a large heavy pot over medium-high heat, melt 2 tbs butter. Add the lobster shells, coarsely chopped carrot, celery, and quartered onion. Sauté, stirring, until shells and vegetables are lightly browned (6–8 minutes). Stir in 2 tbs tomato paste and cook 2 minutes to caramelize.

Deglaze and simmer stock

Pour in 1 cup dry white wine to deglaze, scraping browned bits from the bottom. Reduce wine by half (about 2–3 minutes). Add 8 cups cold water, bay leaf, peppercorns, thyme sprigs, and 1.5 tsp salt. Bring to a gentle simmer, reduce heat to low, and simmer uncovered 30–45 minutes, skimming foam occasionally.

Strain and reduce

Strain the stock through a fine-mesh sieve lined with cheesecloth into a clean pot, pressing gently on solids to extract liquid. Discard solids. If you prefer a more concentrated flavor, return strained stock to the pot and simmer until reduced by about one-third (10–15 minutes). Keep hot.

Soup Base

Sweat aromatics

In a separate heavy pot or Dutch oven, melt 4 tbs butter over medium heat. Add the sliced leek and minced shallots and sweat gently until softened and translucent but not browned, about 6–8 minutes. Add minced garlic and cook 30 seconds until fragrant.

Make roux

Sprinkle 3 tbs all-purpose flour over the softened aromatics and stir to coat. Cook the roux, stirring, 2–3 minutes to remove raw flour taste but do not let it darken much.

Add tomatoes and deglaze

Stir in the diced tomato (or drained canned tomato) and 3 tbs dry sherry (or cognac). Cook 1–2 minutes to blend flavors.

Combine with shell stock

Gradually whisk in about 4 cups of the hot strained lobster stock until smooth, then add the remaining stock. Bring the mixture to a gentle simmer, stirring occasionally. Simmer 15–20 minutes to allow flavors to meld and the roux to thicken the base.

Blend and strain for smooth bisque

Remove from heat. Using an immersion blender, puree the soup until smooth and silky. Alternatively, carefully transfer in batches to a blender and puree. Pass the pureed soup through a fine-mesh sieve or chinois, pressing with the back of a ladle to extract maximum flavor and leave a velvety texture.

Finish with cream and seasoning

Return strained soup to low heat. Stir in 1 cup heavy cream and 1 cup half-and-half (or whole milk) and warm gently—do not boil—until heated through. Add 1 tsp paprika, 1/4 tsp cayenne, and 1/2 tsp freshly ground black pepper. Taste and adjust salt as needed. Add 1 tsp lemon juice to brighten the flavors.

Lobster Meat Finish

Finish lobster meat

In a small skillet over medium heat, melt 1 tbs butter. Add the reserved lobster meat pieces and warm gently for 1–2 minutes until heated through. Season lightly with salt and freshly ground black pepper. Do not overcook to avoid toughness.

Assemble and garnish

Ladle hot bisque into warmed bowls. Spoon the warmed lobster meat pieces into the center of each bowl. Garnish with finely chopped fresh chives and an extra pinch of smoked paprika if desired. Serve immediately with crusty bread or croutons on the side.